OCF and ordinal notation based on a weakly Mahlo cardinal
Original on my site
We will define functions \(\psi, \chi\) by exclusion \(\varphi, \Phi\) in the definition of Rathjen's original functions [1]. That simplifies creation of recursive ordinal notation and system of fundamental sequences. Application of this recursive notation to FGH or extended arrows allows to generate large computable numbers.
Small greek letters \(\alpha, \beta, \gamma, \delta, \xi, \eta\) denote ordinals. Each ordinal \(\alpha\) is identified with the set of its predecessors \(\alpha=\{\beta|\beta
FS for Hypcos's notation (with weakly inaccessibles cardinals) up to Ψ(M^2)
This is the continuation of my previous post. I added few new rules in fundamental sequences system to extend up to \(\psi(I_{I_{I...}})=\psi(M^2)\). That allow to define FS for Hypcos's notation with weakly inaccessibles. I publish post to take into account possible critical remarks and then to add given FS-system in the article List of systems of fundamental sequences.
\(\rho\) and \(\pi\) are always regular cardinals written as \(\Omega_{\nu+1}\) or \(I_{\mu+1}\) i.e. \(\text{cof}(\rho)=\rho\) and \(\text{cof}(\pi)=\pi\).
\(\Omega_\alpha\) with \(\alpha>0\) is the \(\alpha\)-th uncountable cardinal, \(I_\alpha\) with \(\alpha>0\) is the \(\alpha\)-th weakly inaccessible cardinal and for this notation \(I_0=\Omega_0=0\).
Then,
\(C_0(\alpha,\…
Canonical fundamental sequences for Hypcos's notation with the first weakly inaccessible cardinal.
This is the system of fundamental sequences which I propose to use as canonical for Hypcos's notation with the first weakly inaccessible cardinal.
Defenition:
\(\rho\) and \(\pi\) are always regular cardinals i.e. \(\rho,\pi\in\{\Omega_{\nu+1}\}\cup\{ I\}\) i.e. \(\text{cof}(\rho)=\rho\) and \(\text{cof}(\pi)=\pi\).
\( I\) is the first weakly inaccessible cardinal, \(\Omega_\alpha\) with \(\alpha>0\) is the \(\alpha\)-th uncountable cardinal and for this notation \(\Omega_0=0\).
Then,
\(C_0(\alpha,\beta) = \beta\cup\{0,I\} \)
\(C_{n+1}(\alpha,\beta) = \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \cup \{\Omega_\gamma|\gamma\in C_n(\alpha,\beta)\} \cup \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma
Comparison of notations
\(\varepsilon_0\)
Note:
IN denotes I-notation;
EA+CHF denotes Extended Arrows with ordinals \(\alpha\le\varepsilon_0\) written in Cantor normal form; EA+BF denotes Extended Arrows with ordinals \(\alpha\le\Lambda\) written in normal form for Buchholz function whose fundamental sequences are assigned according ruleset for this function;
BAN denotes Bird's array notation;
E^ denotes Cascading-E notation;
xE^ denotes Extended Cascading-E Notation;
Appendix: Extended Arrows's Definition
We define for non-zero natural numbers \(n\), \(b\) and for ordinal number \(\alpha\geq 0\):
1) \(n\uparrow^\alpha b= nb \text{ if }\alpha=0\),
2) \(n\uparrow^{\alpha+1} b = \left\{\begin{array}{lcr} n \text{ if }b=1\\ n\uparrow^{\alpha}(n\uparrow^{\alpha+1} (b-1))\tex…
I-notation
Online calculator for |-notation (up to \(\varepsilon_0\))
Online calculator for |-notation (up to \(\psi_0(\varepsilon_{\Omega_9+1})\))
Expression written in |-notation is exactly equal to output of function of fast growing hierarchy indexed by ordinal number generated by Buchholz's function.
What inspired me:
Chronolegends's Egg Notation
Deedlit's notation
Buchholz's function
Fast-growing hierarchy
a|b corresponds to \(f_b(a)\) where \(f_b\) is a function of fast-growing hierarchy
To the right of the sign "|" :
1) () corresponds to 1, (()) corresponds to \(\omega\) and (...) always corresponds to a countable ordinal number,
2) \(()_b\) corresponds to \(\Omega_b\) where \(\Omega_b=\aleph_b=\psi_b(0)\) denotes b-th uncountable ordinal,
3) \((...)_b\) c…